Perhaps Gus Poyet should have hired a hypnotist to convince his players it was Tuesday or Wednesday or Saturday – any day bar Monday. Sunderland really do not like Mondays and have now gone 12 years and 19 games since winning on one.

This latest failure to perform on the first evening of the working week has left Poyet's side staring at the Championship. Second bottom and four points adrift of safety, Sunderland hold games in hand but face daunting trips to Tottenham – on a Monday – Chelsea, Manchester City and Manchester United as well as a visit from Everton. The mist shrouding Wearside at the final whistle seemed horribly symbolic.

Once again boos rang in Sam Allardyce's ears but this time they were not directed at West Ham's manager – whose team effectively banished lingering relegation fears with this result – but at home players without a league win since 1 February.

"They tried everything but that's what we are," said Poyet. "I'm not going to lie. It hurts, I'm sad. It's not going to be a very nice night for me but we have to keep believing. If the players give up, I'll leave."

Faced with Andy Carroll, Sunderland took the precaution of fielding five at the back but, within 10 minutes, West Ham's "divine ponytail" scored only his second league goal of the season.

As a Newcastle fan, Carroll will have relished connecting with Mark Noble's corner, outjumping John O'Shea before thumping a header beyond Vito Mannone.

"It looked pretty doomy and gloomy in late January," said Allardyce. "But I always said we'd be OK when Andy came back from injury. This outstanding victory has got us out of trouble and almost safe. If Andy gets another two or three goals it'll be difficult for Roy Hodgson to leave him out of England's World Cup squad."

Carroll and company adapted well to the shock of swapping 20C London sunshine for a murky north-eastern six degrees. Even so they were a little lucky when the fallout from James Tomkins' error fell to Lee Cattermole's weaker left foot. Poyet's self-styled enforcer shot straight at Adrián, enabling the goalkeeper to save with his legs before watching Cattermole skew the rebound over the bar.

Generally, though, a West Ham side well schooled in protecting leads and expert at collecting clean sheets, looked confident as Sunderland increasingly seemed to be limping in collective sympathy with their manager. Poyet, nursing a debilitating knee problem, looked on helplessly as his players saw plenty of the ball but were largely reduced to passing it harmlessly between themselves. West Ham had absolutely no inhibitions about pulling 10 men behind the ball, leaving Connor Wickham and Fabio Borini struggling to get behind Allardyce's defence.

Sunderland should have won a penalty when Nolan handled in the area but Howard Webb contentiously ignored their appeals and there was an inescapable sense of another door closing softly on their season.

If some of the half-time jeers were clearly directed at Webb – whose decision was the sort on which careers and livelihoods can hinge – many stemmed from sheer frustration. Lacking natural width without the benched Adam Johnson, Sunderland could not undo a deep-sitting, five-man midfield dominated by the streetwise Kevin Nolan.

West Ham doubled their advantage from another set piece. Tomkins dispatched a free-kick from just inside his own half which Carroll failed to control, leaving the ball spinning free for Mohammed Diamé to attempt a shot. Instead of heading clear, the hapless Santiago Vergini deflected it beyond Mannone. Poyet switched to a back four but Stewart Downing very nearly scored a third for West Ham after dodging Marcos Alonso.

Allardyce's side finally began wobbling when Johnson reduced the deficit after his fellow substitute Craig Gardner nutmegged Pablo Armero. It left the erstwhile England winger to take one touch and send a rising shot curving imperiously into the top corner from the edge of the area. Belatedly dangerous, Sunderland blitzed the visiting box only to be betrayed by their final ball.